Fixed points in non-invariant plane continua

FIXED POINTS IN N ON-INV ARIANT PLANE CONTIN UA ALEXANDER BLOKH AND LEX OVERSTEEGEN Abstract. If f : [ a, b ] → R , with a < b , is contin uous and s uch that a and b are mapped in opp osite directions b y f , then f has a fixed point in I . Suppo se that f : C → C is map and X is a contin uum. W e extend the ab ov e for c e rtain contin uous maps of dendr ites X → D , X ⊂ D and for p ositively oriented maps f : X → C , X ⊂ C with the contin uum X not necessa r ily inv a riant. Then we show that in certain cases a holomorphic map f : C → C must hav e a fixed p oint a in a contin uum X s o that either a ∈ Int ( X ) or f exhibits rotation at a . 1. Intro duction By C w e denote the plane and b y C ∞ the Riemann sphere. The fixed p oin t problem, attributed to [Ste35], is one of the central problems in plane top ology . It can b e form ulated as follow s. fpt Problem 1.1. S upp ose that f : C → C is c ontinuous an d f ( X ) ⊂ X for a non- sep ar ating c ontinuum X . Do es ther e always exi s t a fixe d p oint in X ? As is, Problem 1.1 is not y et solv ed. The most w ell-kno wn particular case f o r whic h it is solve d is that of a map of a closed interv al I = [ a, b ], a < b into itself in whic h case there m ust exist a fixed p oin t in I . In fact, in this case a more general result can b e prov en of whic h the existence of a fixed p oint in an in v ariant interv al is a consequenc e. Namely , instead of considering a map f : I → I consider a map f : I → R suc h that either (a) f ( a ) ≥ a and f ( b ) ≤ b , or (b) f ( a ) ≤ a and f ( b ) ≥ b . Then still there m ust exist a fixed p oint in I whic h is an easy coro llary of the In termediate V alue Theorem applied to the function f ( x ) − x . Observ e that in this case I need not b e in v ariant under f . Observ e a lso that without the a ssumptions on the endp oin ts, the conclusion on the existence of a fixed p oin t inside I cannot b e made b ecause, e.g., a shift map on I do es not hav e fixed p oin ts at a ll. The conditions (a) and (b) ab ov e can be though t of as b oundary conditions impo sing r estrictions on where f maps the b oundary p oin ts of I in R . Our main aim in this pap er is to consider some other cases for whic h Problem 1.1 is solv ed in affirmativ e (i.e., the existence of a fixed p oin t in an in v arian t con t inuum is established) and replace for them the inv arian tness of the con tin uum b y b o undary Date : September 21, 2 008. 2000 Mathematics Su bje ct Classific ation. P rimary 37F10 ; Secondary 37F50 , 37 B45, 37C25 , 54F15, 54H25. Key wor ds and phr ases. Fixe d p oints, plane contin ua, o r iented maps, Complex dynamics; Julia set. The first author was par tially supp or ted by NSF grant DMS-04 5 6748 . The second author was partia lly supp or ted by NSF grant DMS-0405 7 74. 1 2 ALEXANDER BLOKH AND LEX OVERSTEEGEN conditions in the spirit of the a b o v e “in terv al version” of Problem 1.1 . More precisely , instead of assuming that f ( X ) ⊂ X w e will mak e some assumptions on the w ay that f maps p oints of f ( X ) \ X ∩ X . First though let us discuss part icular cases for whic h Problem 1.1 is solv ed. They can b e divided in to tw o categories: either X has sp ecific pro p erties, or f has sp ecific prop erties. The most direct extension of the “interv al particular case” of Problem 1 .1 is, p erhaps, the fo llowing w ell kno wn theorem (see for example[Nad92]). dendr Theorem 1.2. If f : D → D is a c ontinuous map of a dendrite into itself then it has a fi xe d p oint. Here f is just a contin uous map but the con tinuum D is v ery nice. Theorem 1.2 can b e generalized to the case when f maps D in to a dendrite X ⊃ D and certain conditions on to the b eha vior of the p oints of the b o undary of D in X are fulfilled. This presen t s a “non-inv arian t” v ersion of Problem 1.1 for dendrites and can b e done in the spirit of the in terv al case describ ed ab o v e. Moreo ver, with some additional conditions it has consequenc es related to the n umber of p erio dic p oints of f . The details and exact statemen ts can b e found to Section 2 dev oted to the dendrites. Another direction is to consider specific ma ps of the pla ne on a r bit r a ry non- separating contin ua. Let us go ov er kno wn results here. Cart wright and L it tlew o o d [CL51] hav e solved Problem 1.1 in affirmative for orientation preserving homeomor- phisms of the plane. This result w a s generalized to all homeomorphisms b y Bell [Bel78]. The existence of fixed p oin ts for orientation preserving homeomorphisms o f the en tir e plane under v ario us conditions was considered in [Bro12, Bro84, F at87, F ra92, Gui94, FMOT07], and of a p o in t of p erio d t w o for orientation rev ersing home- omorphisms in [Bon04]. The result by Cart wrigh t and Litt lewoo d deals with the case when X is an inv ariant con tinuum. In parallel with the in terv a l case , w e w an t to exte nd this to a larger class of maps of the plane (i.e., not necessarily one-to- one) suc h that certain “b oundary” conditions are satisfied. Our t o ols are mainly based on [FMOT07] and apply to p ositively oriente d maps whic h gene ralize the notion of an orien t a tion preserving homeomorphism (see the precise definitions in Section 2). Our main top ological results are Theorems 3 .1 2 and 3.19. Th e precise conditions in them are somewhat tec hnical - after a ll, we need to describ e “b oundary conditions” of maps of a rbitrary non-separating con tin ua. Ho w ev er it turns out that these conditions ar e satisfied b y holomorphic maps (in pa r ticular, p olynomials), a llo wing us to obta in a few corollaries in this case, essen t ia lly all dealing with the existence of p erio dic p oin t s in certain parts of the Julia set of a p olynomial and degeneracy of certain impressions. 2. Preliminaries prel A map f : X → Y is p e rf e c t if for each compact set K ⊂ Y , f − 1 ( K ) is also compact. A l l m aps c ons i d er e d in this p ap er ar e pr efe ct. Given a con tin uum K ⊂ C , denote by U ∞ ( K ) the comp onen t of C \ K con taining infinity , and by T ( K ) the top olo gic al hul l of K , i.e. the set C \ U ∞ ( K ). By S 1 w e denote the unit circle whic h w e iden tify with R mod Z . FIXED POINTS 3 In this se ction w e will in tro duce a new class of maps whic h are the prop er general- ization of an orientation prese rving homeomorphism. F or complete ness w e will recall some related results fro m [FMOT07] where these maps w ere first introduced. Definition 2.1 (Degree of a map) . Let f : U → C b e a ma p from a simply connected domain U into the plane. L et S b e a p ositiv ely oriented simple closed curv e in U , and w 6∈ f ( S ) b e a p oin t. Define f w : S → S 1 b y f p ( x ) = f ( x ) − w | f ( x ) − w | . Then f w has a well-define d de gr e e (also known as the winding numb er of f | S ab out w ), denoted degree( f w ) = win( f , S, w ). Definition 2.2. A map f : U → C fro m a simply connected domain U is p ositively oriente d (resp ectiv ely , ne gatively oriente d ) prov ided for eac h simple closed curv e S in U and eac h p oin t w ∈ f ( T ( S )) \ f ( S ), degree( f w ) > 0 (degree( f w ) < 0, respectiv ely). A holomor phic map f : C → C is a prototy p e o f a p ositiv ely o rien ted map. Hence the results obtained in this pap er apply to t hem. How ev er, in general p ositiv ely orien ted maps do not hav e to b e light (i.e., a p ositive ly orien ted map can map a sub con tinuum of C t o a p o in t) . Observ e also, that for p oin ts w 6∈ T ( f ( S )) we hav e degree( f w ) = 0. A map f : C → C is orien te d pro vided for each simple closed curv e S and eac h x ∈ T ( S ), f ( x ) ∈ T ( f ( S )). Ev ery p ositiv ely or negativ ely orien ted map is orien ted (indeed, otherwise there exists x ∈ T ( S ) with f ( x ) 6∈ T ( f ( S )) whic h implies that win( f , S, f ( x )) = 0, a contradiction). A map f : C → C is c onfluent provide d for eac h sub con tin uum K ⊂ C and ev ery comp onen t C of f − 1 ( K ), f ( C ) = K . It is w ell kno wn that b oth op en and monotone maps (and hence comp ositions of suc h maps) of con tinua are confluen t. It follows from a result o f Lelek a nd Read [LR74] that each confluen t mapping of the plane is the comp osition o f a monotone map a nd a ligh t op en map. Theorem 2.3 is obtained in [F MOT07]. orient Theorem 2.3. Supp ose that f : C → C is a surje ctive m ap. Then the fol lo wing ar e e quivalent: pnorient (1) f is ei ther p ositively or ne gatively oriente d; iorient (2) f is orien te d ; conf (3) f is c on fluent. Mor e over, if f satisfies these pr op erties then for a n y non-sep ar ating c ontinuum X we have f ( Bd ( X )) ⊃ B d ( f ( X )) . Let X b e an non-separating plane con tinuum. A cr o s s cut of U = C \ X is an op en arc A ⊂ U suc h that Cl ( A ) is an arc with ex actly tw o endp oints in Bd( U ). Eviden tly , a crosscut of U separates U into tw o disjoint domains, exactly one un b ounded. Let S b e a simple closed curv e in C and suppo se g : S → C has no fixed p oin ts on S . Since g has no fixed p oin ts on S , t he p oin t z − g ( z ) is nev er 0. Hence the unit vec tor v ( z ) = g ( z ) − z | g ( z ) − z | alw ays exists. Let z ( t ) b e a con v enien t coun terclo c kwise parameterization o f S b y t ∈ S 1 and define the map v = v ◦ z : S 1 → S 1 b y 4 ALEXANDER BLOKH AND LEX OVERSTEEGEN v ( t ) = v ( z ( t )) = g ( z ( t )) − z ( t ) | g ( z ( t )) − z ( t ) | . Then Ind(g , S), the index of g o n S , is the de gr e e of v . The following theorem (see, e.g., [F MOT07]) is a ma jor to o l in finding fixed p oin ts of con tinuous maps of the plane. basic Theorem 2.4. Supp ose that S ⊂ C is a sim p le close d curve and f : T ( S ) → C is a c ontinuous map such that Ind(f , S) 6 = 0 . Then f ha s a fixe d p oint in T ( S ) . Theorem 2.4 applies to Problem 1.1 as follo ws. Giv en a non-separating con tinuum X ⊂ C o ne constructs a simple closed curv e S approximating X so that the index o f f on S can b e computed. If it is no t equal to zero, it implies the existence of a fixed p oin t in T ( S ), and if S is tight enough, in X . Hence o ur efforts should b e aimed at constructing S and computing Ind(f , S). O ne wa y of doing so is to use Bell’s notion of v ariation whic h w e will now in tro duce. Supp ose that X is a non- separating plane contin uum and S is a simple closed curv e suc h that X ⊂ T ( S ) a nd S ∩ X consists o f mor e than one p oin t. Then we will call S a bumpin g simple close d curve of X . An y subarc of S , b oth of whose endp oints are in X , is called a bumping ar c of X or a link of S . Note that an y bumping arc A of X can b e extended to a bumping simple closed curv e S of X . Hence, ev ery bumping arc has a natural order < inherited from the p ositiv e circular order of a bumping simple closed curv e S con taining A . If a < b are the endp oin ts of A , then we will often write A = [ a, b ]. Also, by the shadow S h ( A ) of A , w e mean the union of all b ounded comp onen t s of C \ ( X ∪ A ). The standar d junction J 0 is the union of the three rays R i = { z ∈ C | z = r e iπ / 2 , r ∈ [0 , ∞ ) } , R + = { z ∈ C | z = r e 0 , r ∈ [0 , ∞ ) } , R − = { z ∈ C | z = r e iπ , r ∈ [0 , ∞ ) } , ha ving the origin 0 in common. By U w e denote the low er half-plane { z ∈ C | z = x + iy , y < 0 } . A junction J v is the image o f J 0 under an y orientation-preserving homeomorphism h : C → C where v = h (0 ). W e will often suppress h and refer to h ( R i ) as R i , and similarly f o r the r emaining rays and the region h ( U ). Definition 2.5 (V ar ia tion on a n arc) . Let f : C → C b e a map, X b e a non- separating plane contin uum, A = [ a, b ] b e a bumping subarc of X with a < b , { f ( a ) , f ( b ) } ⊂ X and f ( A ) ∩ A = ∅ . W e define the variation of f on A wi th r e- sp e ct to X , denoted V ar(f , A), b y the f ollo wing algo r ithm: (1) Cho ose an orientation preserving homeomorphism h of C suc h that h (0) = v ∈ A and X ⊂ h ( U ) ∪ { v } . crossing s (2) Cr ossings: Cons ider the set K = [ a, b ] ∩ f − 1 ( J v ). Mo v e along A fr o m a to b . Eac h time a p oint of [ a, b ] ∩ f − 1 ( R + ) is follow ed immediately b y a p oint of [ a, b ] ∩ f − 1 ( R i ) in K , count + 1. Each t ime a p oin t of [ a, b ] ∩ f − 1 ( R i ) is follo w ed immediately b y a p oin t of [ a, b ] ∩ f − 1 ( R + ) in K , coun t − 1. Coun t no other crossings. (3) The sum of t he crossings found a b o v e is the v ariatio n, denoted V ar(f , A). It is sho wn in [F MOT07] that the v ariatio n do es not dep end on the choice of a junction satisfying the ab o v e listed pro p erties. Informally , one can understand the notion of v ariation as follows. Since f ( A ) ∩ A = ∅ , w e can a lw ays complete A with FIXED POINTS 5 another arc B (now connecting b to a ) to a simple closed curv e S disjoint from J v so that v 6∈ f ( S ). Then it is easy to see that win( f , S, v ) cab b e obtained by summing up V ar(f , A) and the similar coun t for the arc B (observ e that the latter is not the v a r ia tion of B b ecause to compute that w e will need to use another junction “based” at a p oin t of B ). An y partition A = { a 0 < a 1 < · · · < a n < a n +1 = a 0 } ⊂ X ∩ S of a bumping simple close d curv e S of a non-separating con tin uum X suc h that for all i , f ( a i ) ∈ X and f ([ a i , a i +1 ]) ∩ [ a i , a i +1 ] = ∅ is called an al lowable p artition of S . W e will call the bumping arcs [ a i , a i +1 ] link s (of S ) . It is sho wn in [FMOT07] that the v ar iation of a bumping arc is w ell-defined. Moreo ver, it follows from that pap er (see Theorem 2.12 and Remark 2.19) that: FMOT Theorem 2.6. L et S b e a s i m ple close d curve, X = T ( S ) and l e t a 0 < a 1 < · · · < a n < a 0 = a n +1 b e p oints in S (in the p ositive cir cular or der ar ound S ) such that for e ach i , f ( a i ) ∈ T ( S ) a nd, if Q i = [ a i , a i +1 ] , then f ( Q i ) ∩ Q i = ∅ . Then Ind(f , S) = X i V ar(f , Q i ) + 1 . Observ e that if the p oints a i , i = 1 , . . . , n satisfying the prop erties of Theorem 2.6 can b e c hosen then there are no fixed p oints of f in S and Ind(f , S) is well-define d. Theorem 2.6 shows that if w e define V ar(f , S) = P i V ar(f , Q i ), then V ar(f , S) is w ell defined a nd indep enden t of the choice of the allow a ble partition of S a nd of the c hoice of the junctions used to compute V ar(f , Q i ). closed Lemma 2.7. L et f : C → C b e a map, X b e a non-sep ar ating c ontinuum and C = [ a, b ] b e a bumping ar c of X with a < b . L et v ∈ [ a, b ] b e a p oint and let J v b e a junction such that J v ∩ ( X ∪ C ) = { v } . Supp ose that J v ∩ f ( X ) = ∅ . Then ther e exists an ar c I such that S = I ∪ C is a bumping si m ple close d c urve of X and f ( I ) ∩ J v = ∅ . Pr o of. Since f ( X ) ∩ J v = ∅ , it is clear that there exists an a rc I with endpoints a and b near X suc h that I ∪ C is a simple closed curv e, X ⊂ T ( I ∪ C ) and f ( I ) ∩ J v = ∅ . This completes the pro of.  The next corollary giv es a sufficien t condition for the non-negativity o f the v a riation of an a rc. posvar Corollary 2.8. Supp ose f : C → C is a p osi tive l y oriente d map, C = [ a, b ] is a bumping ar c of X , f ( C ) ∩ C = ∅ and J v ⊂ [ C \ X ] ∪ { v } is a junction with J v ∩ C = { v } . Supp ose that f ( { a, b } ) ⊂ X and ther e exists a c o ntinuum K ⊂ X such that f ( K ) ∩ J v ⊂ { v } . Then V ar(f , C) ≥ 0 . Pr o of. Observ e that since f ( a ) , f ( b ) ∈ X then V ar(f , C) is well-defin ed. Cons ider a few cases. Supp ose first that f ( K ) ∩ J v = ∅ . Then, b y Lemma 2 .7 , there exists an a rc I suc h that S = I ∪ C is a bumping simple closed curv e around K and f ( I ) ∩ J v = ∅ (it suffices to c ho ose I v ery close to K ). The n v ∈ C \ f ( S ). He nce V ar(f , C) = win( f , S, v ) ≥ 0. Supp ose next that f ( K ) ∩ J v = { v } . Then we can p erturb the junction J v sligh tly in a small neigh b orho o d of v , obtaining a new junction J d suc h that in tersections of f ( C ) with J v and J d are the same (and, hence, yield the same v ariation) and f ( K ) ∩ J d = ∅ . No w pro ceed as in the first case.  6 ALEXANDER BLOKH AND LEX OVERSTEEGEN 3. Main res ul ts 3.1. Dendrites. In this subsection we generalize Theorem 1.2 to non- inv ariant den- drites. W e will also sho w that in certain cases the dendrite m ust con tain infinitely man y p erio dic cutp oin ts (recall that if Y is a con tin uum and x ∈ Y then v al Y ( x ) is the n um b er o f connected comp onen ts of Y \ { x } , and x is said to b e an endp oint (of Y ) if v al Y ( x ) = 1, a cutp oint (of Y ) if v al Y ( x ) > 1 and a vertex/b r anchp oint (of Y ) if v a l Y ( x ) > 2). These results ha v e applications in complex dynamics [BCO08]. In this subsection giv en t w o p oin ts a, b of a dend rite w e denote b y [ a, b ] , ( a, b ] , [ a, b ) , ( a, b ) the unique closed, semi-op en and op en arcs connecting a and b in the dendrite. More sp ecifically , unless otherwise sp ecified, the situation considered in this subsection is as fo llo ws: D 1 ⊂ D 2 are dendrites and f : D 1 → D 2 is a con tin uous map. Set E = D 2 \ D 1 ∩ D 1 . In other words, E consists of p oints at whic h D 2 “grow s” out o f D 1 . A p oint e ∈ E ma y b e a n endp oin t of D 1 (then there is a unique comp o nent of D 2 \ { e } whic h meets D 1 ) or a cutp oint of D 1 (then there are sev eral comp onen ts of D 2 \ { e } whic h meet D 1 ). The following theorem is a simple extension of the real result claiming t hat if t here are p oints a < b in R such that f ( a ) < a, f ( b ) > b then there exists a fixed p oin t c ∈ ( a, b ) (case (b) described in In tro duction). fixpt-1 Prop osition 3.1. Supp ose that a, b ∈ D 1 ar e such that a sep a r a tes f ( a ) fr o m b and b sep ar ates f ( b ) fr om a . Then ther e e x i s ts a fixe d p oi n t c ∈ ( a, b ) which is a cutp oint of D 1 (and henc e D 2 ). In p articular if ther e a r e two p oints e 1 6 = e 2 ∈ E such that f ( e i ) b elongs to a c omp o n ent of D 2 \ { e i } disjoint fr om D 1 then ther e exists a fixe d p oint c ∈ ( a, b ) which is a cutp oint o f D 1 (and henc e D 2 ). Pr o of. It follow s that w e can find a sequence of p o ints a − 1 , . . . in ( a, b ) suc h tha t f ( a − n − 1 ) = a − n and a − n − 1 separates a − n from b . Cle arly , lim n →∞ a − n = c ∈ [ a, b ] is a fixed p oin t as desired. If there are tw o p oints e 1 6 = e 2 ∈ E suc h tha t f ( e i ) b elongs to a comp onen t o f D 2 \ { e i } disjoin t f r om D 1 then the ab ov e applies to them.  The other real case (case (a)) described in Introduction is somewhat more difficult to generalize. Definition 3.2 extends it (i.e. the real case when a < b are p oints of R suc h that f ( a ) > a and f ( b ) < b ) on to dendrites. bouscr Definition 3.2. Supp ose that in the ab ov e situation the map f is suc h that for eac h non-fixed p oin t e ∈ E , f ( e ) is contained in a comp onen t of D 2 \ { e } whic h meets D 1 . Then we say that f has the b ounda ry scr ambling pr op erty or that it scr ambles the b oundary . Observ e that if D 1 is in v aria n t then f automatically scram bles the b oundary . The next definition presen ts a useful top ological v ersion o f rep elling at a fixed p oin t. wkrep Definition 3.3. Supp ose that a ∈ D 1 is a fixed p oint and that there exists a com- p onen t B of D 1 \ { a } suc h tha t arbitrarily close to a in B there exist fixed cutp oints of f or p oints x separating p from f ( x ). Then sa y that a is a we akly r ep el ling fixe d p oint (of f in B ) . A p erio dic p oin t a is said to b e we a k ly r ep el ling if t here exists n and a comp onen t B of D 1 \ { a } suc h that a is a w eakly rep elling fixed p oin t of f n in B . FIXED POINTS 7 It is easy to see that a fixed p o int a is w eakly rep elling in B if and only if either a is a limit of fixed cutp oints of f in B , or there exist a neigh b orho o d U of a in B and a p oin t x ∈ U \ { a } suc h that U contains no fixed points but a a nd x separates p f r om f ( x ). Indee d, in the latt er case by con tinuit y there exists a p oin t x 1 ∈ ( a, x ) suc h that f ( x − 1 ) = x and this sequenc e of preimages can b e extended to w a r ds a inside ( a, x ) so that it conv erges to a (otherwise it w ould con v erge to a fixed point inside U distinct from a , a con tradiction). In particular, if a is a we ak l y r ep el l i n g fixe d p oint of f then a is a we ak l y r ep el ling fixe d p oint of f n for any n . Moreo v er, since there are only coun tably many v ertices o f D 2 and their images under f and its p o w ers, w e can c ho ose x and its bac kw ard orbit conv erging to a so that all its p oints are cutp o ints of D 2 of v alence 2. F rom no w on w e assume that to eac h w eakly rep elling fixed p oin t a of f in B whic h is not a limit p o int o f fixed cutp o in ts in B w e asso ciate a p oint x a = x ∈ B of v alence 2 in B separating a from f ( x ) and a neigh b orho o d U a = U ⊂ B whic h is the comp onent of B \ { x } con ta ining a . As an imp o rtan t to ol we will need t he follo wing retraction closely related to the described ab o v e situation. retr Definition 3.4. F or eac h x ∈ D 2 there exists a unique a r c (p ossibly a p oin t) [ x, d x ] suc h that [ x, d x ] ∩ D 1 = { d x } . Hence there exists a natural monotone retraction r : D 2 → D 1 defined by r ( x ) = d x , and the ma p g = g f = r ◦ f : D 1 → D 1 whic h is a con tin uous map of D 1 in to itself. W e call the map r the natur al r etr action (of D 2 onto D 1 ) and the map g the r etr acte d (versi o n of ) f . The map g is designed to mak e D 1 in v ariant so that Theorem 1.2 applies to g and allo ws us to conclude that there are g -fixed p o in ts. Ho w eve r these p oin ts a re no t necessarily fixed p oints of f . Indeed, g ( x ) = x means that r ◦ f ( x ) = x . Hence it really means tha t f maps x to a p oin t b elonging to a comp onen t of D 2 \ D 1 whic h “grow s” o ut of D 1 at x . In pa rticular, it means that x ∈ E . Th us, if g ( x ) = x and x / ∈ E then f ( x ) = x . In general, it fo llo ws from the construction that if f ( x ) 6 = g ( x ), then g ( x ) ∈ E b ecause p oints of E are exactly those p oints of D 1 to which p oints of D 2 \ D 1 map under r . W e are ready to pro v e our first lemma in this direction. fixpt0 Prop osition 3.5. Supp ose that f scr am bles the b oundary. Then f has a fixe d p oin t. Pr o of. W e ma y assume that there are no f -fixed p oin ts e ∈ E . By Theorem 1.2 the map g f = g has a fixed p o in t p ∈ D 1 . It follow s from the f a ct tha t f scramble s the b oundary that p oin ts of E are not g -fixed. Hence p / ∈ E , a nd by the argumen t right b efore the lemma f ( p ) = p as desired.  It f o llo ws fr o m Prop osition 3.1 and Prop osition 3.5 that the o nly b eha vior of p oints in E whic h do es not force the existence of a fixed p oint in D 1 is when exactly one p oin t e ∈ E maps into a comp onen t of D 2 \ { e } whic h is disjoin t f r o m D 1 whereas an y other p oint e ′ ∈ E maps into a component of D 2 \ { e ′ } whic h is not disjoin t fr o m D 1 . The next lemma shows that in some cases p can b e c hosen to b e a cutp o in t of D 1 . fxctpt Lemma 3.6. Supp ose that f s cr ambles the b ounda ry and al l f -fixe d endp oints of D 1 ar e we akly r ep e l ling. Then ther e is a fixe d cutp o i n t of f . Pr o of. Supp ose tha t f has no fixed cutp oints. By Prop osition 3.5, the set of fixed p oin ts of f is not empt y . Hence w e may assume that al l fixed p oints o f f are endp oin ts 8 ALEXANDER BLOKH AND LEX OVERSTEEGEN of D 1 . Let a, b b e distinct fixed p oin ts of f . Then it follows that either U a ⊂ U b , or U b ⊂ U a , or U a ∩ U b = ∅ . Indeed, supp ose that x a ∈ U b . Let us sho w t hat then U a ⊂ U b . Indeed, otherwise by Lemma 3.5.(1) there exists a fixed p oin t c ∈ ( x a , x b ), a con tra diction. No w, supp ose that neither x a ∈ U b nor x b ∈ U a . Then clearly U a ∩ U b = ∅ . Consider an op en cov ering of the set of a ll fixed p oin ts a ∈ D 1 b y their neigh b orho o ds U a and c ho ose a finite sub co v er. By the ab ov e w e may assume that its consists of pa ir wise disjoint sets U a 1 , . . . , U a k . Consider now the comp onent Q o f D 1 whose endp oints are the p o ints a 1 , . . . , a k . It is easy to see that f | Q , with Q considered as a sub dendrite of D 2 , scram bles the b oundary and has no fixed endp oints . Hence an f -fixed p oin t p ∈ Q , exis ting b y Lemma 3.5, m ust b e a cutp o int of D 1 .  Lemma 3.6 is helpful in the next theorem which sho ws that under some rather w eak assumptions on p erio dic p oin ts the map has infinitely many p erio dic cutp o in ts. infprpt Theorem 3.7. Supp ose that f : D → D is c ontinuous and al l its p erio dic p oints ar e we akly r ep el ling. Then f has infinitely man y p erio dic cutp oints. Pr o of. By w ay o f contradiction supp ose that there are finitely many p erio dic cutp oints of f . Without loss of g eneralit y w e ma y assume that these are p oints a 1 , . . . , a k eac h of whic h is fixe d under f . Let A = ∪ k i =1 a i and let B b e comp onent of D \ A . Then B is a subdendrite of D to whic h the ab o v e to ols apply: D plays t he role of D 2 , B pla ys the role of D 1 , and E is exactly the b oundary Bd( B ) of B (by the construction Bd( B ) ⊂ A ). Supp ose that each p oint a ∈ Bd( B ) is we akly rep elling in B . Then by the assumptions o f the theorem Lemma 3.6 applies to this situation. It follows that there exists a fixed cutp oint of B , a con tradiction. Hence for some a ∈ Bd B we ha v e that a is not w eakly rep elling in B . Since b y the assumptions a is we akly rep elling, there exists another comp o nent, sa y , C , of D \ A suc h that a ∈ Bd( C ) and a is weakly rep elling in C . W e can no w apply the same argument to C . If all b oundary p oin ts of C a re w eakly rep elling in C then b y Lemma 3 .6 C will contain a fixed cutp oint, a con tr a diction. Hence t here exists a p oin t d ∈ A suc h that d is not w eakly rep elling in C and a comp onen t F of D \ A whose closure meets C at d , and d is weakly rep elling in F . Clearly , af ter finitely man y steps this pro cess will ha v e to end ultimately leading to a comp onen t Z of D \ A suc h that all p oin ts of Bd( Z ) are w eakly rep elling in Z . Since here the set Bd( Z ) pla ys the role of the set E from ab o v e and b y the assumptions of the theorem we see that Lemma 3.6 applies to Z a nd there exists a fixed cutp oin t of Z , a con tradiction.  An imp or tan t application of Theorem 3.7 is to the dendritic top olo gic al Julia sets . They can b e defined as follows. Consider an eq uiv alence relation ∼ on the unit circle S 1 . Equiv alence classes of ∼ will b e called ( ∼ -)cla s s es and will b e denoted b y b o ldf a ce letters. A ∼ -class consisting o f t w o p oin ts is called a l e af ; a class consisting o f at least three p oin ts is called a gap (this is more restrictiv e than Th urston’s definition in [Thu85]; we follo w [BL02] in our presen tation). Fix an integer d > 1 and denote the map z d : S 1 → S 1 b y σ d . Then ∼ is said to b e a ( d -)invarian t lamination if: (E1) ∼ is close d : the g raph of ∼ is a closed set in S 1 × S 1 ; (E2) ∼ defines a lamina tion , i.e., it is unlinke d : if g 1 and g 2 are distinct ∼ -classes, then their con vex hulls Ch( g 1 ) , Ch( g 2 ) in the unit disk D are disjoint, (D1) ∼ is forwar d invariant : for a class g , the set σ d ( g ) is a class to o FIXED POINTS 9 whic h implies that (D2) ∼ is b ac k war d invariant : for a class g , its preimage σ − 1 d ( g ) = { x ∈ S 1 : σ d ( x ) ∈ g } is a union of classes; (D3) for any gap g , the map σ d | g : g → σ d ( g ) is a c overing map with p ositive orientation , i.e., for ev ery connected comp onen t ( s, t ) of S 1 \ g the arc ( σ d ( s ) , σ d ( t )) is a connected comp onent of S 1 \ σ d ( g ) . The lamination in which all p oin ts of S 1 are equiv alent is said to de gener ate . It is easy to see t ha t if a f orw a r d inv ariant lamination ∼ has a class with non-empt y in te- rior then ∼ is degenerate. Hence equiv alence classes of an y non- degenerate forw a rd in v ariant lamination are totally disconnected. Call a class g critic al if σ d | g : g → σ ( g ) is not one-to-one, (pr e)critic al if σ j d ( g ) is critical for some j ≥ 0, and (pr e)p erio d ic if σ i d ( g ) = σ j d ( g ) for some 0 ≤ i < j . A gap g is wandering if g is neither (pre)p erio dic nor (pre)critical. Let p : S 1 → J ∼ = S 1 / ∼ b e the quotien t map of S 1 on to its quotien t space J ∼ , let f ∼ : J ∼ → J ∼ b e the map induced b y σ d . W e call J ∼ a top olo gic al Julia set and the induced map f ∼ a top olo gic al p olynomial . It is easy to see that if g is a ∼ -class then v al J ∼ ( p ( g )) = | g | whe re b y | A | w e denote the cardinality of a set A . lamwkrp Theorem 3.8. Supp ose that the top olo gic al Julia set J ∼ is a den drite and f ∼ : J ∼ → J ∼ is a top olo gic al p olynomial. Then al l p erio dic p oin ts of f ∼ ar e we akly r ep el l i n g and f ∼ has infinitely many p erio dic cutp oints. Pr o of. Supp ose that x is an f ∼ -fixed p o in t and set g = p − 1 ( x ). If x is an endp o in t then g is a singleton. Connect x with a p oint y 6 = x . Then the arc [ x, y ] ⊂ J ∼ con ta ins po in ts y k → x of v alence 2 b ecause, as is w ell kno wn, t here are no more than coun tably many vertice s of J ∼ . It follo ws that ∼ -classes p 1 ( y k ) ar e lea ve s separating g fro m the rest of the circle and rep elled from g under t he action of σ . Hence f ∼ ( y i ) is separated from x b y y i and so x is w eakly rep elling. Supp ose that x is not an endp oint. C ho ose a v ery small connected neighbor ho o d U of x . It is easy to see that eac h comp onent A of U \ { x } corresp onds to a unique c hord ℓ A ∈ Bd(Ch( g )). Moreo v er, fo r eac h comp onent A of U \ { x } there exists a unique comp onent B of U \ { x } suc h t ha t f ∼ ( A ) ∩ B 6 = ∅ . Hence there is a map h from the set A of all comp onen ts of U \ { x } to itself. Supp ose that there exists E ∈ A and n > 0 suc h that h n ( E ) = E . Then it follows that the endp oints of ℓ E are fixed under σ n . Connect x with a p oint y ∈ E and ch o ose, as in the previous paragraph, a sequence of p o ints y k ∈ [ x, y ] , y k → x of v a lence 2. Then again by the rep elling prop erties o f σ n it follo ws that f ∼ ( y i ) is separated from x by y i and so x is weakly rep elling (f or f m ∼ in E ). It remains to sho w that there E ∈ A with h n ( E ) = E for some n > 0 m ust exist. Supp ose otherwise. T o eac h comp o nen t C of U \ { x } w e asso ciate t he corresp onding comp onen t J C of J ∼ \ { a } containing C . Then there are only finitely many suc h comp onen t s C o f U \ { x } t ha t J C con ta ins a critical p o int; denote their collection b y C . Let us show that an ev entual h -image of ev ery E ∈ A mus t coincide with an elemen t of C . Indeed, otherwise there is a comp onen t E ∈ A such that all h k ( E ) are distinct and the map f ∼ | J h k ( E ) is a homeomorphism. Clearly , this implies the existence of a w andering subcontin uum K of J ∼ . Ho wev er by Theorem C [BL02] this is imp ossible. 10 ALEXANDER BLOKH AND LEX OVERSTEEGEN Hence all p erio dic p oints of f ∼ are w eakly rep elling and b y Theorem 3.7 f ∼ has infinitely many p erio dic cutp oin ts.  3.2. Posi tiv ely oriented maps of the plane. In t his subsection we will first obta in a general fixed p oint theorem whic h shows t ha t if a non-separating plane con tinuum, not necessarily in v a rian t, maps in an appropriat e w a y , then it m ust con ta in a fixed p oin t. This extends Theorem 3.9. Let us denote the family of a ll p ositiv ely or ien ted maps of the plane b y P . fmot1 Theorem 3.9 ([FMOT07]) . Supp ose that f ∈ P and that X ⊂ C i s a non- s e p ar ating c ontinuum such that f ( X ) ⊂ X . T h en ther e exis ts a fixe d p o i n t p ∈ X . T o pro ceed we will need to generalize Corollary 2.8 to a more general situation. T o this end we intro duce a definition similar to the one giv en f or dendrites in the previous section. scracon Definition 3.10. Supp ose that f ∈ P and X is non-separating con tin uum. Supp ose that there exist n ≥ 0 disjoint non- separating con tin ua Z i suc h that the following prop erties hold: (1) f ( X ) \ X ⊂ ∪ i Z i ; (2) for a ll i , Z i ∩ X = K i is a non-separating contin uum; (3) for a ll i , f ( K i ) ∩ [ Z i \ K i ] = ∅ . Then the ma p f is said to scr amble the b o undary (of X ). If instead of (3 ) w e ha ve (3a) for all i , either f ( K i ) ⊂ K i , or f ( K i ) ∩ Z i = ∅ then we say that f str ongly scr ambles the b oundary (of X ) . In either case, K i ’s are called exit c ontinua (of X ) . Note that since Z i ∩ X is a con tin uum, X ∪ ( S Z i ) is a non- separating con tinuum . Sp eaking of maps whic h (strongly) scram ble the b oundary , w e alw ays use the notation from this definition unless explicitly stated otherwise. Observ e that in the situation o f Definition 3.10 if X is inv ariant t hen f auto- matically strongly scram bles the b oundary b ecause we can simply take the set of exit contin ua to b e empt y . Also, if f strongly scram bles the b oundary of X and f ( K i ) 6⊂ K i for a n y i , t hen it is easy to see that there exists ε > 0 suc h that for ev ery p oin t x ∈ X either d ( x, Z i ) > ε , or d ( f ( x ) , Z i ) > ε . Let us now prov e the follo wing tec hnical lemma. posvar+ Lemma 3.11. Supp ose that f : C → C scr amb l e s the b oundary of X . L et Q b e a bumping ar c of X with endp oints a < b ∈ X such that f ( { a, b } ) ⊂ X and f ( Q ) ∩ Q = ∅ . Then V ar(f , Q) ≥ 0 . Pr o of. W e will use the notation as sp ecified in the lemma. Supp ose first that Q \ S Z i 6 = ∅ and choo se v ∈ Q \ S Z i . Since v ∈ Q \ S Z i and X ∪ ( S Z i ) is non- separating, there exists a junction J v , with v ∈ Q , suc h that J v ∩ [ X ∪ Q ∪ S Z i ] = { v } and, hence, J v ∩ f ( X ) ⊂ { v } . No w the desired result follows from Corollary 2.8. Observ e that if Q \ ∪ Z i = ∅ then Q ⊂ Z i for some i and so Q ∩ X ⊂ K i . In particular, b ot h endp oin ts a, b of Q are con t a ined in K i . Cho ose a p oin t v ∈ Q . Then again there is a junction connecting v and infinity outside X (except p ossibly for v ). Since a ll sets Z j , j 6 = i are p o sitively distant f r o m v and X ∪ ( S i 6 = j Z i ) is non-separating, the junction J v can be c hosen to a v oid all sets Z j , j 6 = i . Now, b y (3) f ( K i ) ∩ J v ⊂ { v } , hence by Lemma 2.8 V ar(f , Q) ≥ 0.  FIXED POINTS 11 Lemma 3.11 is applied in Theorem 3.1 2 in whic h w e sho w that a map whic h stro ngly scram bles the b oundary has fixed p o in ts. In fact, it is a ma jor tec hnical to ol in o ur other results to o. Indeed, if we can construct a bumping simple closed curv e S around X suc h that the endp o ints of its links map back into X while these links mo ve completely off themselv es, the lemma w ould imply that the v ariation of S is non-negativ e. By Theorem 2.6 this w ould imply t hat the index o f S is p ositiv e. Hence b y Theorem 2.4 there are fixed points in T ( S ). Cho osing S to b e sufficien tly tight w e see that there are fixed p oints in X . F or the sake of con venie nce w e no w sk etc h the pro of of Theorem 3.1 2 whic h allow s us to emphasize the main ideas rather than details. The main steps in constructing S a re as follows. First w e assume by w ay of contradiction that a map f : C → C has no fixed p oints in X . Then by Theorem 3.9 it implies that f ( X ) 6⊂ X and that f ( K i ) 6⊂ K i for an y i . By the definition of strong scrambling then f ( K i ) is far a w ay from Z i for an y i . No w, since there are no fixed p oints in X we can choo se the links in S to b e v ery small so that they will all mo ve off themselv es. Ho w ev er some of them will ha v e endp oin ts mapping outside X whic h prev en ts us from directly applying Lemma 3.11 to them. These links will b e enlarg ed b y concatenating them so that the images of the endp oints of these concatenations are inside X and these concatenations still map off themselv es. The bumping simple closed curv e S then remains as b efore, ho w ev er the represen tation of S as the union of links c hanges b ecause w e enlarge some of them. Still, the construction shows that Lemma 3.11 applies t o the new “bigg er” links and as b efo r e this implies the existence of a fixed p oint in X . T o ac hieve the goal of replacing some links in S by t heir concatenations we consider the links whic h are mapp ed outside X in detail using t he fact that f strongly scramble s the b oundar y (indeed, all other links ar e suc h that Lemma 3 .1 1 already applies to them). The idea is to consider the links of S whose concatenation is a connected piece of S mapping into one Z i . Then if w e b egin the concatenation right b efore the images of links en ter Z i and stop it right after the ima g es of the links exit Z i w e will ha ve one condition of Lemma 3.11 satisfied b ecause the endp oints of the thus constructed new “big” concatenation link T of S map into X . W e no w need to v erify that T mo v es off itself under f . Indeed, this is easy to see for the end-links of T : each end-link has t he image “crossing” into Z i from X \ Z i , henc e the images of end-links are close to K i . Ho we v er the sets K i are ma pp ed far a w ay from Z i b y the definition of strong scram bling and b ecause none of K j ’s is in v aria n t b y the assumption. This implies that the end-links themselv es m ust b e far aw a y from Z i . If no w w e mo v e from link to link inside T we see that those links cannot approa ch Z i to o closely b ecause if they do they will hav e to “cross o v er K i ” into Z i , and then their images will hav e to b e close to the ima g e of K i whic h is far aw a y fr o m Z i , a con tra diction with the fact that all links in T ha v e endp oin ts whic h map in to Z i . In other w ords, the dynamics of K i prev en ts the new bigger links from getting ev en close to Z i under f whic h sho ws that they mo v e off themselv es as desired. As b efore, w e no w apply Theorem 2.6 to see that Ind(f , S) = V ar(f , S) + 1 and then Theorem 2.4 to see that this implies the existence of a fixed p oin t in X . Giv en a compact set K denote by B ( K, ε ) the op en set of all p oin ts whose distance to K is less than ε . 12 ALEXANDER BLOKH AND LEX OVERSTEEGEN fixpt Theorem 3.12. Supp ose that f : C → C str ongly s c r ambles the b oundary of X . Then f has a fixe d p oint in X . Pr o of. If f ( X ) ⊂ X then the result follo ws from [FMOT07]. Similarly , if there exists i suc h that f ( K i ) ⊂ K i , then f ha s a fixed p oint in T ( K i ) ⊂ X and w e are also done. Hence we ma y assume f ( X ) \ X 6 = ∅ and f ( K i ) ∩ Z i = ∅ for all i . Supp ose that f is fixed p oin t free. Then there exists ε > 0 suc h that for all x ∈ X , d ( x, f ( x )) > ε . W e ma y assume that ε < min { d ( Z i , Z j ) | i 6 = j } . W e now choose constan ts η ′ , η , δ and a bumping simple closed curv e S of X so t hat the following holds. (1) 0 < η ′ < η < δ < ε/ 3. (2) F or each x ∈ X ∩ B ( K i , 3 δ ), d ( f ( x ) , Z i ) > 3 δ . (3) F or each x ∈ X \ B ( K i , 3 δ ), d ( x, Z i ) > 3 η . (4) F or eac h i there exists a p oint x i ∈ X suc h that f ( x i ) = z i ∈ Z i and d ( z i , X ) > 3 η . (5) X ⊂ T ( S ) and A = X ∩ S = { a 0 < a 1 < · · · < a n < a n +1 = a 0 } in the p ositiv e circular order around S . (6) f | T ( S ) is fixed p oin t free. (7) F or the closure Q i = [ a i , a i +1 ] of a comp onen t of S \ X , diam( Q )+diam( f ( Q )) < η . (8) F or any t wo p oints x, y ∈ X with d ( x, y ) < η ′ w e ha v e d ( f ( x ) , f ( y )) < η . (9) A is an η ′ -net in Bd( X ). Observ e that b y t he triangle inequality , Q i ∩ f ( Q i ) = ∅ for every i . Claim 1. T her e exists a p oint a j such that f ( a j ) ∈ X \ ∪ B ( Z i , η ) . Pr o of of Claim 1 . Set B ( Z i , 3 η ) = T i and show that there exists a p oin t x ∈ Bd( X ) with f ( x ) ∈ X \ ∪ T i . Indeed, supp ose first that n = 1 . Then f ( K 1 ) ⊂ X \ T 1 and w e can choo se any p oint of K 1 ∩ Bd( X ) as x . No w, supp ose that n ≥ 2. Observ e that the sets T i are pairwise disjoint compacta. By Theorem 2.3 f (Bd( X )) ⊃ Bd( f ( X )). Hence there are p oints x 1 6 = x 2 in Bd( X ) suc h that f ( x 1 ) ∈ Z 1 ⊂ T 1 , f ( x 2 ) ∈ Z 2 ⊂ T 2 . Since the sets f − 1 ( T i ) ∩ X are pairwise disjoin t non-empt y compacta w e see that the set V = Bd( X ) \ ∪ f − 1 ( T i ) is non- empt y (b ecause Bd( X ) is a con t inuum). Now w e can c ho ose an y p oint of V as x . It remains to notice that b y the c hoice of A we can find a p oin t a j suc h that d ( a j , x ) < η ′ whic h implies that d ( f ( a j ) , f ( x )) < η and hence f ( a j ) ∈ X \ ∪ B ( Z i , η ) as desired.  There exists a p oin t x 1 suc h t hat f ( x 1 ) = z 1 is more than 3 η - distan t from X . W e can find a ∈ A suc h t ha t d ( a, x 1 ) < η ′ and hence f ( a ) 6∈ X is at least 2 η -distan t from X . On the o ther hand, by Claim 1 there are p oin ts o f A mapp ed in t o X . Let < denote the circular order on the set { 0 , 1 , . . . , n + 1 } defined b y i < j if a i < a j in the p ositiv e circular o rder around S . Then w e can find n (1) < m (1) suc h that the follo wing claims hold. (1) f ( a n (1) − 1 ) ∈ X \ ∪ Z i . (2) f ( a r ) ∈ f ( X ) \ X for all r with n (1) ≤ r ≤ m (1) − 1 (and so, since diam( f ( Q u )) < ε/ 3 for a ny u and d ( Z s , Z t ) > ε for all s 6 = t , there exists i (1) with f ( a r ) ∈ Z i (1) for all n (1) ≤ r < m (1)). (3) f ( a m (1) ∈ X \ ∪ Z i . FIXED POINTS 13 Consider the a r c Q ′ 1 = [ a n (1) − 1 , a m (1) ] ⊂ S and sho w that f ( Q ′ 1 ) ∩ Q ′ 1 = ∅ . As we w alk along Q ′ 1 , w e b egin outside Z i (1) at f ( a n (1) − 1 ), then enter Z i (1) and w alk insid e it, and then exit Z i (1) at f ( a m (1) ). Since ev ery step in this walk is rather short (diam( Q i ) + diam( f ( Q i )) < η ), we see that d ( f ( a n (1) − 1 ) , Z i (1) ) < η and d ( f ( a m (1) ) , Z i (1) ) < η . On the ot her hand for each r , n (1) ≤ r < m (1) w e hav e f ( a r ) ∈ Z i (1) , hence w e see that d ( f ( a r ) , Z i (1) ) < η for each r , n (1) ≤ r < m (1). This implies that d ( a r , K i (1) ) > 3 δ (b ecause otherwise f ( a r ) w o uld b e far t her a wa y from Z i (1) ) and so d ( a r ) , Z i (1) ) > 3 η ( b ecause a r ∈ X \ B ( K i (1) , 3 δ )). Since diam( Q ) + diam( f ( Q )) < η , then d ( Q ′ 1 , Z i (1) ) > 2 δ > 2 η . On the ot her hand, d ( f ( a r ) , Z i (1) ) < η similarly implies that d ( f ( Q ′ 1 ) , Z i (1) ) < 2 η . Thus indeed f ( Q ′ 1 ) ∩ Q ′ 1 = ∅ . This allo ws us to replace the original division of S into its prime links Q 1 , . . . , Q n b y a new o ne in whic h Q ′ 1 pla ys the role o f a new prime link; in o t her w ords, w e simply delete the p oin ts { a n (1) , . . . , a m (1) − 1 } from A . Contin uing in t he same manner and mo ving a long S , in the end w e obtain a finite set A ′ = { a 0 = a ′ 0 < a ′ 1 < · · · < a ′ k } suc h that for each i w e hav e f ( a ′ i ) ∈ X ⊂ T ( S ) and for each arc Q ′ i = [ a ′ i , a ′ i +1 ] we ha v e f ( Q ′ i ) ∩ Q ′ i = ∅ . Hence, by Theorem 2 .6, Ind(f , S) = P Q ′ i V ar(f , Q ′ i ) + 1. Since b y Lemma 3.11, V ar(f , Q ′ i ) ≥ 0 for all i , Ind(f , S) ≥ 1 con tra dicting the fact that f is fixed p oint free in T ( S ).  3.3. Maps with isolated fixed p oints. Now we consider maps f ∈ P with isolated fixed p oints; denote t he set of suc h maps by P i . W e need a few definitions. crit Definition 3.13. Giv en a map f : X → Y w e sa y that c ∈ X is a critic al p oint of f if for any neighborho o d U of c , there exist x 1 6 = x 2 ∈ U suc h t ha t f ( x 1 ) = f ( x 2 ). Hence, if x is not a critical p oin t of f , then f is lo cally one-t o -one near x . If a p o in t x b elongs to a contin uum collapsed under f then x is critical; also any p oin t whic h is an a ccumulation p oin t of collapsing con tinua is critical. How ev er in this case t he map around x may b e mono t o ne. A more intere sting case is when the map a round x is not monoto ne; then x is a br anch p oint of f and it is critical ev en if there a r e no collapsing con tin ua close b y . O ne can define the lo c al de gr e e deg f ( a ) as t he n umber of comp onen ts of f − 1 ( y ), for a p oint y close to f ( a ), whic h are non- disjoin t from a small neigh b orho o d of a . Then branchpoints are exactly the p oints at whic h the lo cal degree is more tha n 1. Notice that since w e do not assume an y smo othness, a critical p oin t ma y w ell b e b oth fixed (p erio dic) and be suc h that sm all neigh b orho o ds of c = f ( c ) map ov er themselv es by f . The next definition is closely related to that of the index of t he map on a simple closed curv e. indpt Definition 3.14. Supp ose that x is a fixed p oint of a map f ∈ P i . Then the lo c al index of f at x , denoted b y Ind(f , x), is defined as Ind(f , S) where S is a small simple closed curv e around x . It is easy to see that if f ∈ P i , t hen the lo cal index is w ell-defined, i.e. do es not dep end on the choice of S . By mo difying a translation map one can giv e an example of a homeomorphism of the plane whic h has exactly o ne fixed p oin t x with lo cal index 0. Still in some cases the lo cal index at a fixed p oin t mus t b e p ositiv e. toprepat Definition 3.15. A fixed p o in t x is said to b e top olo gic al ly r ep el ling if there exists a sequence of simple closed curv es S i → { x } suc h that x ∈ in t( T ( S i )) ⊂ T ( S i ) ⊂ 14 ALEXANDER BLOKH AND LEX OVERSTEEGEN in t( T ( f ( S i )). A fixed p oin t x is said to b e top olo gic a l ly a ttr acting if there exists a sequence of simple closed curv es S i → { x } not con taining x and suc h tha t x ∈ in t( T ( f ( S i )) ⊂ T ( f ( S i )) ⊂ in t ( T ( S i )). ind1 Lemma 3.16. If a is a top o l o gic al ly r ep el ling fixe d p oint then Ind(f , a) = deg f ( a ) ≥ 1 wher e d is the lo c al de gr e e. If howe v e r a is a top olo gic al ly attr acting fixe d p oint then Ind(f , a) = 1 . Pr o of. Consider the case of the rep elling fixed p oin t a . Then it follows that, as x runs along a small simple closed curv e S with a ∈ T ( S ), the ve ctor from x to f ( x ) pro duces the same winding n um b er as the v ector from a to f ( x ), and it is easy to see that the latter equals deg f ( a ). The ar gumen t with attracting fixed p oint is similar.  If how eve r x is neither to p ologically rep elling nor top olo gically attracting, then Ind(f , x) could b e greater than 1 ev en in the non-critical case. Indeed, tak e a neutral fixed p oin t of a rational function. Then it follo ws that if f ′ ( x ) 6 = 1 then Ind(f , x) = 1 while if f ′ ( x ) = 1 then Ind(f , x) is the multiplicit y at x (i.e., the lo cal degree of the map f ( z ) − z at x ). This is related the following useful theorem. It is a vers ion a more general, top olo g ical ar g ume nt principle stated in the conv enien t for us form. argupr Theorem 3.17. Supp ose that f ∈ P i . Then for a n y simple close d curve S ⊂ C which c ontains no fixe d p oints of f its index e quals the sum of lo c al indic es taken over al l fixe d p oints in T ( S ) . Theorem 3.17 implies The orem 2.4 but pro vides more information. In particular if S we re a simple closed curv e and if we knew that the lo cal index at any fixed p oint a ∈ T ( S ) is 1, it would imply that Ind(f , S) equals the n umber n ( f , S ) of fixed p o in ts of f in T ( S ). By the ab ov e analysis this holds if all f -fixed p oints in T ( S ) are either rep elling, or attracting, or neutral a nd such that f has a complex deriv ative f ′ in a small neighborho o d of x , and f ′ ( x ) 6 = 1. In the spirit of the previous parts of the pap er, w e are still concerned with find- ing f -fixed p oints inside non-in v arian t con tinua of whic h f (stro ng ly) scramble s the b oundary . Ho wev er w e no w sp ecify the t yp es of fixed p oints w e are lo o king fo r . Th us, the main result of this subsection prov es the existence of sp ecific fixed p o in ts in non- degenerate con t inua satisfying the appropriate b oundary conditions and sho ws that in some cases suc h con tin ua m ust b e degenerate. It is in this latter form that w e apply the result later on in Section 4. Giv en a non-separating con tin uum X , a ra y R ⊂ C \ X from ∞ whic h lands on x ∈ X (i.e., R \ R = { x } ) and a crosscut Q of X w e sa y that Q a nd R cr oss esse n tial ly pro vided there exists r ∈ R suc h that the subarc [ x, r ] ⊂ R is con tained in Sh( Q ). The next definition complemen ts the previous one. repout Definition 3.18. If f ( p ) = p and p ∈ Bd( X ) then we sa y that f r ep els outside X at p provid ed there exists a ray R ⊂ C \ X from ∞ whic h lands on p and a sequence of simple closed curv es S j b ounding closed disks D j suc h that D 1 ⊃ D 2 ⊃ . . . , ∩ D j = { p } , f ( D 1 ∩ X ) ⊂ X , f ( S j \ X ) ∩ D j = ∅ and for eac h j there exists a comp onen t Q j of S j \ X suc h that Q j ∩ R 6 = ∅ and V ar(f , Q j ) 6 = 0. If f ∈ P and f scram bles the b o undary of X , then by Lemma 3.11, for an y comp onen t Q of S j \ X w e ha v e V ar(f , Q) ≥ 0 so that in this case V ar(f , Q j ) > 0. FIXED POINTS 15 The next theorem is the main result of this subsection. locrot Theorem 3.19. Supp ose that f ∈ P i , and X ⊂ C is a non-sep ar ating c ontinuum or a p oint. Mor e over, the fol lowing c o n ditions hold. fV (1) F or e ach fixe d p oint p ∈ X we have that Ind(f , x) = 1 and f r ep els outside X at p . (2) The map f scr ambl e s the b ounda ry of X . Mor e over, either f ( K i ) ∩ Z i = ∅ , or ther e exists a neighb orho o d W i of K i with f ( W i ∩ X ) ⊂ X . Then X is a p oint. Pr o of. Supp ose that X is not a p o in t. Since f ∈ P i , there exists a simply connected neigh b orho o d V of X suc h that all fixed p oin ts { p 1 , . . . , p m } of f | V are contained in X . W e will show that then f m ust ha ve at least m + 1 fixe d po ints in V , a con t radiction. The pro o f will pro ceed lik e t he pro of of The orem 3.12: w e construct a tigh t bumping simple closed curve S suc h tha t X ⊂ T ( S ) ⊂ V . W e will show t hat for an appropriate S , V ar(f , S) ≥ m . Hence Ind(f , S) = V ar(f , S) + 1 ≥ m + 1 and by Theorem 3.17 f m ust ha v e at least m + 1 fixed p oints in V . Let us c ho ose neigh b orho o ds U i of exit con tinua K i satisfying conditions listed b elo w. (1) F or n 1 < i ≤ n by assumption (2) of the theorem w e may assume that f ( U i ∩ X ) ⊂ X . (2) F or 1 ≤ i ≤ n 1 w e ma y assume that d ( U i ∪ Z i , f ( U i )) > 0. (3) W e may assume tha t T ( X ∪ S U i ) ⊂ V and U i ∩ U k = ∅ for all i 6 = k . (4) W e ma y assume that ev ery fixed p oint of f con ta ined in U i is con tained in K i . Let { p 1 , . . . , p t } b e all fixed p oints of f in X \ S i K i and let { p t +1 , . . . , p m } b e all the fixed p oin ts con ta ined in S K i . Observ e that then by the c hoice of neigh b orho o ds U i w e hav e p i ∈ X \ ∪ U s if 1 ≤ i ≤ t . Also, it fo llo ws that fo r eac h j, t + 1 ≤ j ≤ n there exists a unique r j , n 1 < r j ≤ n suc h that p j ∈ K r j . F or eac h fixed p o int p j ∈ X c ho ose a ray R j ⊂ C \ X la nding o n p j , a s sp ecified in Definition 3.18, and a small simple closed curv e S j b ounding a closed disk D j suc h that the follo wing claims ho ld. (1) D i ∩ R j = ∅ for all i 6 = j , (2) f ( S j \ X ) ∩ D j = ∅ . (3) T ( X ∪ S j D j ) ⊂ V . (4) [ D j ∪ f ( D j )] ∩ [ D k ∪ f ( D k )] = ∅ fo r all j 6 = k . (5) f ( D j ∩ X ) ⊂ X . (6) Denote b y Q ( j, s ) the comp onen t s of S j \ X ; then there exists Q ( j, s ( j )) , a comp onen t of S j \ X , with V ar(f , Q(j , s(j))) > 0 and Q ( j, s ( j )) ∩ R j 6 = ∅ . (7) [ D j ∪ f ( D j )] ∩ S U i = ∅ for all 1 ≤ j ≤ t . (8) If t < j ≤ n then [ D j ∪ f ( D j )] ⊂ U r j . Note tha t b y (1) for all i 6 = j , Sh( Q ( j, s ( j )) ∩ Q ( i, s ( i )) = ∅ . W e need to c ho ose a few constants . First choose ε > 0 suc h that for all x ∈ X \ S D j , d ( x, f ( x )) > 3 ε . Then b y con tin uit y w e can c ho ose η > 0 suc h t ha t for each set H ⊂ V of diameter less than η w e hav e diam( H ) + diam( f ( H )) < ε and for eac h crosscut C of X disjoin t from ∪ D j w e hav e that f ( C ) is disjoin t from C (observ e that outside ∪ D j all p oints of X mo ve by a b o unded aw a y from zero distance). Finally w e c ho ose δ > 0 so that the following inequalities hold: 16 ALEXANDER BLOKH AND LEX OVERSTEEGEN (1) 3 δ < ε , (2) 3 δ < d ( Z i , Z j ) for a ll i 6 = j , eU (3) 3 δ < d ( Z i , [ X ∪ f ( X )] \ [ Z i ∪ U i ]), (4) 3 δ < d ( K i , C \ U i ), (5) if f ( K i ) ∩ Z i = ∅ , then 3 δ < d ( f ( U i ) , Z i ∪ U i ). Also, giv en a crosscut C w e can asso ciate to its endp oin ts external angles α , β whose ra ys land a t these endp oints from the appropriate side of X determined b y the lo cation of C (so that the op en region o f the plane enclosed by a tigh t equip otential b et w een R α and R β , the se gmen ts of the r a ys from the equip oten tial to the endpo in ts of C , a nd C itself, is disjoin t fr om X ). Thus w e can talk ab out the angula r measure of Q ( j, s ( j )); denote b y β the minim um of all suc h angular measures tak en ov er a ll crosscuts Q ( j, s ( j )). No w, ch o ose a bumping simple closed curv e S ′ of X whic h satisfies t wo conditions: all its links are (a) less tha n δ in diameter, and (b) are of angular measures less than β . Clearly this is p ossible. Then w e amend S ′ as follows. Let us follo w S ′ in the p ositiv e direction starting at a link outside ∪ D j . Then at some moment for the first time we will b e w alking along a link of S ′ whic h enters some D j . As it happ ens, the link L ′ of S ′ in tersects some Q ( j, s ) with endp oints a, b and en ters the shado w Sh( Q ( j, s )). Later o n w e will b e walkin g outside Sh ( Q ( j, s )) mov ing along some link L ′′ . In this case w e replace the en tire segmen t of S ′ from L ′ to L ′′ b y three links: the first one is a deformation of L ′ whic h has the same initial endp oint as L ′ and the terminal p oin t as a , then Q ( j, s ), and then a deformat io n of L ′′ whic h b egins at b and ends at the same terminal p oin t as L ′′ . In this w a y we mak e sure t ha t for all crosscuts Q ( j, s ) either they are links of S ′ or t hey are con tained in the shado w of a link o f S ′ . Moreov er, b y the c hoice of β crosscuts Q ( j, s ( j )) will ha ve to b ecome links of our new bumping simple closed curv e S . By the c ho ice o f η and b y the prop erties of crosscuts Q ( j, s ) it fo llows that an y link of S is disjoint from its image, and fo r eac h j , Q ( j, s ( j )) ⊂ S and V ar(f , Q(j , s(j))) > 0 . W e w ant to compute the v ariation of S . Eac h link Q ( j, s ( j )) con tributes at least 1 to w ards V ar(f , S), and w e wan t to sho w that all ot her links hav e non-negativ e v aria- tion. T o do so w e w ant to apply Lemma 3.11. Hence w e need to v erify tw o conditions on a crosscut listed in Lemma 3 .11. O ne of them fo llo ws from the previous paragraph: all links of S mov e off themselv es. Ho w eve r the other condition of Lemma 3 .11 ma y not b e satisfied by some links o f S b ecause some of their endp oin ts ma y map off X . T o ensure that for our bumping simple closed curv e endp oints e of its links map back in to X w e ha v e to enlarge links o f S and replace some of them b y their concatenations (this is similar to what was done in Theorem 3.12). Then w e will hav e to c hec k if the new “bigger” links still ha v e images disjoin t from themselv es. Supp ose that X ∩ S = A = { a 0 < a 1 < · · · < a n } and a 0 ∈ A is suc h that f ( a 0 ) ∈ X (t he argumen ts similar to those in Theorem 3 .12 sho w that w e can may this assumption without lo ss of generalit y). Let t ′ b e minimal suc h tha t f ( a t ′ ) 6∈ X and t ′′ > t ′ b e minimal suc h that f ( a t ′′ ) ∈ X . Then f ( a t ′ ) ∈ Z i for some i . Denote b y [ a l , a r ] a subarc of S with the endp oin ts a r and a l and mo ving from a l to a r is the p ositiv e direction. Since ev ery comp onen t o f [ a t ′ , a t ′′ ] \ X has diameter less t han δ , f ( a t ) ∈ Z i \ X for all t ′ ≤ t < t ′′ . Moreov er, for t ′ ≤ t < t ′′ , a t 6∈ U i . T o see this note that if f ( K i ) ∩ Z i = ∅ , then by the ab ov e made c hoices f ( U i ) ∩ Z i = ∅ , and if FIXED POINTS 17 f ( K i ) ∩ Z i 6 = ∅ , then f ( U i ∩ X ) ⊂ X by the assumption. Hence it follows from the prop ert y (3) of the constant δ that f ([ a t ′ − 1 , a t ′′ ]) ∩ [ a t ′ − 1 , a t ′′ ] = ∅ and w e can remov e the p oints a t , f o r t ′ ≤ t < t ′′ from the pa rtition A of S . By con tin uing in the same fashion we o bta in a subset A ′ ⊂ A suc h that for the closure of eac h comp onen t C of S \ A ′ , f ( C ) ∩ C = ∅ a nd for b oth endp oin ts a a nd a ′ of C , { f ( a ) , f ( a ′ ) } ⊂ X . Moreo ver, for eac h j , Q ( j, j ( s )) is a comp onen t o f S \ A ′ . No w w e can apply a v ariat io n of t he standard arg umen t sk etc hed in Section 2 aft er Theorem 2 .4 and applied in the pro of of Theorem 3.12; in this v aria tion instead of Theorem 2.4 w e use the fact t ha t f satisfies the a rgumen t principle. Indeed, b y Theorem 2.6 and Lemma 3.11, Ind(f , S) ≥ P j V ar(f , Q(j , j(s))) + 1 ≥ m + 1 and by the Theorem 3.17 f has a t least m + 1 fixed p oin t s in T ( S ) ⊂ V , a contradiction.  Theorem 3.1 9 implies the following degenera te Corollary 3.20. Supp os e that f and a no n-de gener ate X satisfy al l the c onditions state d in The or em 3.19. Then either f do e s not r ep el outside X at one of its fixe d p oints, or the lo c al inde x at one of its fixe d p oints is not e qual to 1 . The last lemma of this section gives a sufficien t and v erifiable condition for a fixed p oin t a b elonging to a lo cally inv arian t contin uum X to b e suc h t ha t the map f r ep els outside X ; w e apply the lemma in t he next section. repel Lemma 3.21. Supp ose that f : C → C is p ositively oriente d, X ⊂ C is a c ontinuum and p is a fixe d p oint of f such that: (1) ther e exists a neighb orho o d U of p such that f | U is one-to-one and f ( U ∩ X ) ⊂ X , (2) ther e exists a close d disk D ⊂ U c ontaining p in its interior such that f ( ∂ D ) ∩ D = ∅ and ∂ D \ X has at le ast two c omp onents, (3) ther e exists a r ay R ⊂ S \ X f r o m infinity such that R = R ∪ { p } , f | R : R → R is a ho m e omorphism and for e ach x ∈ R , f ( x ) s ep ar ates x fr om ∞ in R . Then ther e exis ts a c omp onent C of ∂ D \ X so that C ∩ R 6 = ∅ , V ar(f , C) = +1 and f r ep els outside X at p . Pr o of. W e ma y assume that X \ U c on tains a con t inuum. Let D ∞ = S \ D be the op en disk at infinity and let ϕ : D ∞ → S \ X b e a conformal map suc h t hat ϕ ( ∞ ) = ∞ . Then T = ϕ − 1 ( R ) is a ray in D ∞ whic h compactifies on a p oin t b p ∈ S 1 . Let Q j b e all comp onen ts of ϕ − 1 ( ∂ D \ X ). Then eac h Q j is a crosscut of D ∞ . Let O = { z ∈ D ∞ | f ◦ ϕ ( z ) ∈ S \ X and define F : O → D ∞ b y F ( z ) = ϕ − 1 ◦ f ◦ ϕ ( z ). Note that T ∪ S Q j ⊂ O . W e ma y assume that Q 1 separates b p from ∞ in D ∞ and that no o ther Q j separates Q 1 from ∞ in D ∞ . Claim. F ( Q 1 ) separates Q 1 from ∞ in D ∞ . Pr o of of Claim . Let T ∞ b e the comp onent of T \ Q 1 whic h con ta ins ∞ and let T p b e the comp onent of T \ Q 1 whic h con ta ins b p . Let a = T ∞ ∩ Q 1 and b = T p ∩ Q 1 . Cho ose a p oin t b ′ ∈ T p v ery close to b so that the subarc [ b, b ′ ] ⊂ T p is contained in ϕ − 1 ( D ). Let T ′ p ⊂ T p b e the closed subarc from b p to b ′ . Cho o se an op en arc A in the b ounded comp onen t of D ∞ \ Q 1 , v ery close to Q 1 from a t o the p o in t b ′ ∈ T p so that f | T ′ p ∩ A ∪ T ∞ is one to-one. Put Z = T ′ p ∩ A ∪ T ∞ , then Q 1 ∩ Z = { a } a nd F ( Q 1 ) ∩ F ( Z ) = { F ( a ) } . Since F is a lo cal orientation preserving homeomorphism 18 ALEXANDER BLOKH AND LEX OVERSTEEGEN near a , F ( Z ) ente rs the b o unded comp onen t o f D ∞ \ F ( Q 1 ) at F ( a ) and neve r exits this comp o nent aft er entering it. Moreov er, if q is an endp o in t of Q 1 , then p oints very close to q o n Q 1 and their images are on the same side of T . Since F ( a ) separates a from ∞ on Z and an initial segmen t of T p (with endp oint b p ) is contained in F ( Z ), b p ∈ Sh ( F ( Q 1 )). This completes the pro of o f the claim. Let us compute the v ariation V ar(F , Q 1 ) of the crosscut Q 1 with resp ect to the con tinuum S 1 . Since the computatio n is indep enden t of the c hoice of the Junction [FMOT07], w e can choose a junction J v with junction p oint v ∈ Q 1 so that eac h of the three rays R + , R i and R − in tersect F ( Q 1 ) in exactly one p oint. Hence V ar(F , Q 1 ) = +1. Since ϕ is an orien tation preserving homeomorphism, V ar(f , ϕ (Q 1 )) = +1 and w e are do ne.  4. App lica tions The results in the previous section can b e used to obtain results in complex dy- namics (see for example [BCO08]). W e will sho w that in certain cases con tinu a (e.g., impressions of external r ays) must b e degenerate. Supp ose that P : C → C is a complex p olynomial o f degree d with a connected Julia set J . Let the filled-in Julia set b e denoted by K = T ( J ). W e denote the external ra ys of K by R α . It is w ell kno wn [DH85 a] that if t he degree of P is d and σ : C → C is defined by σ ( z ) = z d , then P ( R α ) = R σ ( α ) . Let for λ ∈ C , L λ : C → C b e defined by L λ ( z ) = λz . Supp ose that p is a fixed p oint in J and λ = f ′ ( p ) with | λ | > 1 (i.e., p is a r ep el ling fixed p oint). Then there exists neigh b orho o ds U ⊂ V of p and a conformal isomorphism ϕ : V → D suc h that fo r all z ∈ U , P ( z ) = ϕ − 1 ◦ L λ ◦ ϕ ( z ). Now, a fixed p oint p is p ar ab olic if P ′ ( p ) = e 2 π ir for some rational n umber r ∈ Q . A nice description of the lo cal dynamics at a parab olic fixed p oint can b e found in [Mil00]. If p is a rep elling or parab olic fixed p oint then [DH85a] there exist k ≥ 1 external ra ys R α ( i ) suc h tha t σ | { α (1) ,...,α ( k ) } : { α (1) , . . . , α ( k ) } → { α (1) , . . . , α ( k ) } is a p ermu- tation, P ( R α ( i ) ) = R σ ( α ( i )) , for each j , R α ( j ) lands on p and no other external ra ys land on p . Also, if P ( R α ( i ) ) = R α ( i ) for some i , then σ ( α ( j )) = α ( j ) for all j . It is kno wn that t wo distinct external rays are not homotopic in the complemen t of K . Giv en an external r a y R α of K w e denote by Π( α ) = R α \ R α the princi p l e set of α , and b y Imp ( α ) the impr ess i on of α ( see [Mil00]). G iv en a set A ⊂ S 1 , w e extend the ab ov e notatio n b y Π( A ) = S α ∈ A Π( α ) and Imp( A ) = S α ∈ A Imp( α ). Let X ⊂ K b e a non-separating contin uum or a p oint suc h that: P1 (1) P air wise disjoint non-separating con tin ua/ p oin ts E 1 ⊂ X, . . . , E m ⊂ X and finite sets of angles A 1 = { α 1 1 , . . . , α 1 i 1 } , . . . , A m = { α m 1 , . . . , α m i m } are giv en with i k ≥ 2 , 1 ≤ k ≤ m . P2 (2) W e hav e Π( A j ) ⊂ E j (so the set E j ∪ ( ∪ i j k =1 R α j k ) = E ′ j is closed and connected). P3 (3) X in tersects a unique comp onent C of C \ ∪ E ′ j , suc h that X \ S E j = C ∩ K . W e call suc h X a gener al puzzle-pie c e a nd call the con tinua E i the e xit c ontinua of X . Observ e that if U is a F ato u domain then either a general puzzle-piece X con ta ins U , o r it is disjoint from U . F or eac h j , the set E ′ j divides the plane in to i j op en sets whic h w e will call we d ges (at E j ) ; denote by W j the w edge whic h contains X \ E j . FIXED POINTS 19 Let us no w consider the condition (1) of Theorem 3.19. It is easy to see tha t applied “as is” to the p olynomial P at parab olic p oints it is actually not true. Indeed, as explained ab o v e the lo cal index at parab olic fixed p oin ts at whic h the deriv ative equals 1 is greater than 1. And indeed, in our case there are fixed rays la nding at all fixed p oints, therefore [Mil00] the deriv ativ es at all the parab olic p oints in X are equal to 1. The idea whic h allo ws us to solv e this problem is that w e can c ha nge our map P inside t he parab olic domains in question without compromising the rest of the ar g umen ts and making these parab olic p oin ts top ologically rep elling. The th us constructed new map g will satisfy conditions of Theorem 3.1 9 . pararepe l Lemma 4.1. Supp ose that X is a c ontinuum and p ∈ X is a p ar ab olic p oint of a p olynomial f and R is a fixe d external which lands at p . Then f r ep els outside X at p . Pr o of. Let p ∈ X b e a pa rab olic fixed p oint and let F i b e the parab o lic domains con ta ining p in their b oundaries B i . Since there are fixed ra ys landing at p , a ll F i ’s are forward inv arian t. By a nice recen t result of Yin a nd Ro esc h [R Y08], the b oundary B i of eac h F i is a simple closed curve and f | B i is conjugate to the map z → z d ( i ) for some d ( i ) ≥ 2. Let ψ : F i → D b e a conforma l isomorphism. Since Bd( F i ) is a simple closed curv e, ψ extends to a ho meomorphism. Since f | B i is conjugate to the map z → z d ( i ) , it now f o llo ws t ha t the map P | F i can b e replaced b y a map top olo gically conjugate by ψ to the map g i ( z ) = z d ( i ) on the closed unit disk. Let g b e t he map defined b y g ( z ) = P ( z ) for eac h z ∈ C \ S F i and g ( z ) = g i ( z ) when z ∈ F i . Then g is clearly a p ositiv ely orien ted map. The w ell-kno wn analysis of the dynamics of P around parab olic p oin ts [Mil00 ] implies that P rep els p oin ts aw a y from p outside par a b olic domains F i . In other w ords, w e can find a sequence of simple closed curve s S i whic h satisfy conditions of Definition 3.15 and show that p is a top ologically rep elling p oint of g . Hence the lo cal index Ind(g , p) at p equals 1. On the other hand, by Lemma 3.2 1 and prop erties of X it follows that g rep els outside X at p . Since f and g coincide out side X , f a lso rep els at p .  The fo llo wing corollary follo ws fro m Theorem 3 .19. pointdyn Corollary 4.2. S upp o se that X ⊂ K is a non-sep a r ating c ontinuum o r a p oint. Th en the fol lowing c laims hold for X . 1 (1) Supp ose that X is a gener al puzzle-pie c e with exit c o ntinua E 1 , . . . , E m such that either P ( E i ) ⊂ W i , or E i is a fixe d p oint. I f X do es no t c ontain an invariant p ar ab olic dom ain, al l fixe d p oin ts which b elong to X ar e r ep el ling or p ar ab olic, and al l r ays landing at them ar e fixe d, then X is a r ep el ling or p ar ab olic fixe d p o i n t. 2 (2) Supp ose that X ⊂ J i s an i n variant c ontinuum, al l fix e d p oints which b elong to X ar e r ep el ling or p ar ab oli c , and al l r ays landi n g at them ar e fixe d. Then X is a r ep el ling or p ar ab olic fixe d p oint. Pr o of. By w ay o f contradiction w e can assume that X is not a p oin t. Let us show that no pa r a b olic domain with a fixed p oint o n its b oundary can in tersect X . Indeed, in the case (2 ) X ⊂ J and no F atou domain in tersects X , so there is not hing t o pro v e. In t he case (1) observ e that since X is a g eneral puzzle-piec e, it has to con t a in 20 ALEXANDER BLOKH AND LEX OVERSTEEGEN the closure of the en tire par a b olic domain with a fixed p oint, sa y , p on its b oundary . Then the f act that all external rays landing at p are fixed implies that all par a b olic domains containing p in their b oundar ies are in v a r ia n t. Since b y the assumptions X con ta ins no inv arian t parab olic domain, it do es not con tain any of them. So, X is disjoin t from all para b olic domains con taining a fixed p oin t in their b o undaries. T o apply Theorem 3.19 w e need to ve rify that its conditions apply . It is easier to c hec k t he conditio n (2) first. T o do so, observ e first that f ( X ) ∩ X 6 = ∅ . Indeed, otherwise no set E i is a fixed p oint and f ( X ) m ust b e con tained in one of the w edges formed b y some E ′ l but not in the w edge W l . This implies that E l neither is a fixed p oin t, nor is mapp ed in W l , a con tradiction. Th us, f ( X ) ∩ X 6 = ∅ and w e can think of f ( X ) as a con tin uum whic h “gr o ws” out of X . No w, any comp onent of f ( X ) \ X whic h in tersects E k for some k mus t b e con ta ined in one of the we dges at E k , but not in W k . T ak e the closure of their union and then its top o logical h ull union E i and denote it b y Z i . It is easy to c heck no w tha t with these sets Z i the map P scrambles the b oundary of X . Moreo v er, if E i is mapp ed in to W i then clearly P ( E i ) ∩ Z i = ∅ (b ecause Z i is con tained in the other wedges at E i but is disjoint from W i ). On the other hand, if E i is a fixed p oint then it is a rep elling or parab olic fixed p o int with a few external fixed r ays landing at it . Hen ce in a small neigh b orho o d U i of E i the in tersection U i ∩ X maps in to X as desired in the condition (2) o f Theorem 3.19. By Lemmas 3.2 1 and 4.1 P rep els o ut side X at an y fixed p oin t in X . Moreo v er, using the map g constructed in the pro of of Lemma 4.1, w e see that g is top ologically rep elling at p and, hence Ind(g , p) = +1. Hence the conditions of Theorem 3 .19 are satisfied for the map g . Th us, b y Theorem 3 .1 9 we conclude that X is a p oint as desired.  The fo llo wing is an immediate corollary o f Theorem 4.2. rot-neut r Corollary 4.3. Supp ose that for a non-s e p ar ating no n- degenerate c ontinuum X ⊂ K one of the fol lowing f a cts hold. (1) X is a gener al puzzle-pie c e with exit c ontinua E 1 , . . . , E m such that either P ( E i ) ⊂ W i , or E i is a fix e d p oint. (2) X ⊂ J is an inva ri a nt c ontinuum. Then either X c ontains a non-r ep el ling and non-p ar ab olic fixe d p oint, or X c ontains an invariant p ar ab olic domain, or X c ontains a r ep el ling or p ar ab olic fixe d p oint at which a non- fi xe d r ay lands . Finally , the follo wing corollary is useful in pro ving the degeneracy o f certain im- pressions and establishing lo cal contin uit y of the Julia set at some p oints. Corollary 4.4. L et P : C → C b e a c omplex p olynomial a n d R α is a fixe d external r ay la nding on a r ep el ling or p a r a b o lic fixe d p oint p ∈ J . Supp o s e that T ( Imp ( α )) c ontains only r ep el li n g or p ar ab o l i c p erio dic p oints. Then Im p ( α ) is d e g ener ate. Pr o of. Let X = Imp ( α ). Since R α is a fixed external r ay , P ( X ) ⊂ X . Clearly P do es not rotate at p . Supp ose that p ′ is another fixed p o in t of P in X and R β is an external ra y landing at p ′ . Then P ( R β ) also lands on p ′ . If P rota t es at p ′ , then p ′ is a cut p oint of X . This w ould contradict the fa ct that X = Imp( α ). Hence P do es not rotate at any fixed p oint in X and the result follows from Corollary 4.2.  FIXED POINTS 21 Reference s [Aki99] V. Akis, On the plane fixe d p oint pr oblem , T op olog y P r o c. 24 (199 9), 15–31 . [Bel67] H. Bell, On fix e d p oint pr op erties of plane c ontinua , T rans. A. M. S. 128 (196 7), 539 – 548. 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